magnetization field at criticality in the ising model
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Magnetization field at criticality in the Ising model Christophe Garban ENS Lyon and CNRS Federico Camia (Vrije Universiteit Amsterdam) Joint work with Charles Newman (NYU) and Percolation and Interacting Systems, MSRI, February 2012 C. Garban

  1. Magnetization field at criticality in the Ising model Christophe Garban ENS Lyon and CNRS Federico Camia (Vrije Universiteit Amsterdam) Joint work with Charles Newman (NYU) and Percolation and Interacting Systems, MSRI, February 2012 C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 1 / 25

  2. C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 2 / 25

  3. C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 2 / 25

  4. What are the fluctuations of the magnetiza- tion field ? C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 2 / 25

  5. N N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 3 / 25

  6. + + − − − − − − + + + + + − − − + + − − − − − − + + − + + + − + + + − − − − + + − + − − + − − − + + − + N − − − − + + + − + − − − + + − + N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 3 / 25

  7. • If the spins are i.i.d � σ x → N (0 , 1) 1 N + + − − − − − − + + + + + − − − + + − − − − − − + + − + + + − + + + − − − − + + − + − − + − − − + + − + N − − − − + + + − + − − − + + − + N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 3 / 25

  8. • If the spins are i.i.d � σ x → N (0 , 1) 1 N � σ x δ x 1 → W , two- + + − − − − − − N + + + + + − − − dimensional white noise + + − − − − − − + + − + + + − + + + − − − − + + − + − − + − − − + + − + N − − − − + + + − + − − − + + − + N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 3 / 25

  9. + + − − − − − − + + + + + − − − + + − − − − − − + + − + + + − + + + − − − − + + − + − − + − − − + + − + N − − − − + + + − + − − − + + − + N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 4 / 25

  10. Ising model To each configuration σ ∈ {− 1 , 1 } N 2 , one + + associates the Hamiltonian − − − − − − + + + + + − − − + + − − − − − − + + − + + + − + + + − − − − H ( σ ) := − � i ∼ j σ i σ j + + − + − − + − − − + + − + N − − − − + + + − + − − − + + − + N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 4 / 25

  11. Ising model To each configuration σ ∈ {− 1 , 1 } N 2 , one + + associates the Hamiltonian − − − − − − + + + + + − − − + + − − − − − − + + − + + + − + + + − − − − H ( σ ) := − � i ∼ j σ i σ j + + − + − − + − − − + + − + N − − − − + + + − And we define: + − − − + + − + P β ( σ ) ∝ e − β H ( σ ) N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 4 / 25

  12. • If β < β c � σ x → N (0 , σ 2 1 β ) N � σ x δ x → σ β W 1 + + − − − − − − N + + + + + − − − + + − − − − − − + + − + + + − + + + − − − − + + − + − − + − − − + + − + N − − − − + + + − + − − − + + − + N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 5 / 25

  13. • If β < β c � σ x → N (0 , σ 2 1 β ) N � σ x δ x → σ β W 1 + + − − − − − − N + + + + + − − − + + − − − − − − + + − + + + − + • If β > β c + + − − − − + + − + − − + − − − + + − + N − − − − + + + − + − − − + + − + N N 2 � σ 0 � − N 2 � σ 0 � + C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 5 / 25

  14. • If β < β c ⊕ � σ x → N (0 , σ 2 1 β ) N � σ x δ x → σ β W 1 + + − − − − − − N + + + + + − − − + + − − − − − − + + − + + + − + • If β > β c + + − − − − + + − + − − + − − − � σ x − N 2 E + + − + � � σ 0 N − − − − → N (0 , σ 2 β ) + + + − N + − − − + + − + • As β → β c , σ β ր ∞ N C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 5 / 25

  15. • If β < β c ⊕ � σ x → N (0 , σ 2 1 β ) N � σ x δ x → σ β W 1 + + − − − − − − N + + + + + − − − + + − − − − − − + + − + + + − + • If β > β c + + − − − − + + − + − − + − − − � σ x − N 2 E + + − + � � σ 0 N − − − − → N (0 , σ 2 β ) + + + − N + − − − + + − + • As β → β c , σ β ր ∞ N And now: what about β = β c ?? C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 5 / 25

  16. Which normalization ? To avoid boundary issues, consider our system on the torus Z 2 / N Z 2 . We have: � � � � Var ( σ x ) = σ x σ y E x , y

  17. Which normalization ? To avoid boundary issues, consider our system on the torus Z 2 / N Z 2 . We have: � � � � Var ( σ x ) = σ x σ y E x , y N 2 � � � ≍ σ 0 σ y E y

  18. Which normalization ? To avoid boundary issues, consider our system on the torus Z 2 / N Z 2 . We have: � � � � Var ( σ x ) = σ x σ y E x , y N 2 � � � ≍ σ 0 σ y E y | y | − 1 / 4 (by Onsager) N 2 � ≍ y N 2 N 2 N − 1 / 4 = N 15 ≍ 4

  19. Which normalization ? To avoid boundary issues, consider our system on the torus Z 2 / N Z 2 . We have: � � � � Var ( σ x ) = σ x σ y E x , y N 2 � � � ≍ σ 0 σ y E y | y | − 1 / 4 (by Onsager) N 2 � ≍ y N 2 N 2 N − 1 / 4 = N 15 ≍ 4 � σ x Hence it is natural to look at the random variable m ( N ) := N 15 / 8 . Question Does m ( N ) have a ( unique ) scaling limit ?

  20. “Subtle” issue of renormalization Theorem (McCoy, Wu, 1967) As N → ∞ ∼ c N − 1 / 4 � � σ 0 , 0 σ N , N where c = 2 1 / 12 e 3 ζ ′ ( − 1 ) C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 7 / 25

  21. “Subtle” issue of renormalization Theorem (McCoy, Wu, 1967) As N → ∞ ∼ c N − 1 / 4 � � σ 0 , 0 σ N , N where c = 2 1 / 12 e 3 ζ ′ ( − 1 ) Proposition ( Rotational invariance of the two-point function) � � σ 0 σ x − → � x � 2 →∞ 1 � � σ 0 σ � x � 2 C. Garban (ENS Lyon and CNRS) Magnetization field of the Ising model 7 / 25

  22. Notations / definitions Rescaled lattice a Z 2 , a ≪ 1. Definition (Renormalized magnetization field) + + − − − − − − + + + + + − − − + + − − − − − − Φ a := 15 + + − + + + − + � δ x σ x a 8 + + − − − − + + − + − − x ∈ a Z 2 + − − − + + − + m a := � Φ a , 1 [ 0 , 1 ] 2 � − − − − + + + − + − − − + + − + a Z 2 m a L := � Φ a , 1 [ 0 , L ] 2 � 1

  23. Notations / definitions Rescaled lattice a Z 2 , a ≪ 1. Definition (Renormalized magnetization field) + + − − − − − − + + + + + − − − + + − − − − − − Φ a := 15 + + − + + + − + � δ x σ x a 8 + + − − − − + + − + − − x ∈ a Z 2 + − − − + + − + m a := � Φ a , 1 [ 0 , 1 ] 2 � − − − − + + + − + − − − + + − + a Z 2 m a L := � Φ a , 1 [ 0 , L ] 2 � 1 Question The field Φ a ∈ D ′ . Is it the case that Φ a converges as a → 0 to some random distribution Φ ?

  24. Road map • First part : Theorem (Camia, G., Newman) (i) The magnetization field Φ a on a Z 2 has a unique scaling limit as the mesh a → 0 .

  25. Road map • First part : Theorem (Camia, G., Newman) (i) The magnetization field Φ a on a Z 2 has a unique scaling limit as the mesh a → 0 . (ii) On a finite domain Ω with boundary conditions ξ , the magnetization field | Ω ∩ a Z 2 has a scaling limit Φ ξ Φ a Ω .

  26. Road map • First part : Theorem (Camia, G., Newman) (i) The magnetization field Φ a on a Z 2 has a unique scaling limit as the mesh a → 0 . (ii) On a finite domain Ω with boundary conditions ξ , the magnetization field | Ω ∩ a Z 2 has a scaling limit Φ ξ Φ a Ω . (iii) In particular, m a has a unique limiting law m = m ξ which depends on the boundary conditions (usually + , − , or free ).

  27. Road map • First part : Theorem (Camia, G., Newman) (i) The magnetization field Φ a on a Z 2 has a unique scaling limit as the mesh a → 0 . (ii) On a finite domain Ω with boundary conditions ξ , the magnetization field | Ω ∩ a Z 2 has a scaling limit Φ ξ Φ a Ω . (iii) In particular, m a has a unique limiting law m = m ξ which depends on the boundary conditions (usually + , − , or free ). (iv) The scaling limit is NOT Gaussian.

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